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Research

Motto

``In general, although in one sense I hope that I am actually growing modest about the quaternions, from my seeing so many peeps and vistas into future expansions of their principles, I still must assert that this discovery appears to me to be as important for the middle of the nineteenth century as the discovery of fluxions was for the close of the seventeenth.'' [William Rowan Hamilton in 1853 ten years after his discovery of quaternions]

My main research interest is to understand the global analytical, cohomological and arithmetic properties of complete hyperkähler manifolds (i.e. a manifold with a Riemannian metric, with holonomy preserving a certain quaternionic structure on the tangent bundle) of non-compact type and find exciting applications of this in other parts of mathematics and physics. The zoo of spaces I am intrested in include moduli spaces of instantons on ALE spaces, more generally Nakajima's quiver varieties, toric hyperkähler varieties, moduli spaces of magnetic monopoles on R^3, moduli spaces of Higgs bundles on a Riemann surface; and more generally hyperkähler spaces appearing in the non-Abelian Hodge theory of a curve (like moduli of flat GL(n,C)-connections and character varieties) and in the Geometric Langlands Program.

The cohomological properties I am interested are first to find what analogues of the Hodge-theoretical Kähler package survive for these hyperkähler manifolds and second to find the correct analouges of the Atiyah-Singer index theorems for our hyperkähler manifolds:

  • 1. In the (compact) Kähler case this package contains Hodge theory on L^2 harmonic forms and its consequences like the strong Hard Lefschetz theorem. Thus I am naturally led (e.g. in our joint paper with Hunsicker and Mazzeo) to questions about Hodge theory on hyperkähler manifolds of this non-compact type.
  • 2. In the (compact) manifold case we have the Atiyah-Singer index theorem and all of its consequences in the Kähler case like the Hirzebruch-Riemann-Roch theorem and the Hirzebruch signature theorem. In our non-compact hyperkähler case even an integration theory was missing; which has recently been adressed in a joint paper with Proudfoot. Explicit calculations and complete conjectures of these integrals on the moduli space of Higgs bundles have been pursued in a joint project with Szenes.

    These questions were originally motivated by physics (e.g. in my thesis), in the form of Sen's conjectures and its relatives, but recently some new interesting applications of these problems arose in the theory of Yang-Mills instantons on various gravitational instantons (e.g. in joint papers with Etesi), string theoretical mirror symmetry (e.g. in joint papers with Thaddeus, and in recent joint work with Rodriguez-Villegas), combinatorics of matroids (in joint papers with Sturmfels and Swartz), the arithmetic of some of these hyperkähler spaces and in turn representation theory of finite groups and Lie algebras of Lie type (both in recent joint work with Rodriguez-Villegas and Emmanuel Letellier), representation theory of quivers and Kac-Moody algebras (in my latest project with a proof of a conjecture of Kac from 1982) and recently I am also thinking on applications to knot theory. In all these applications Hodge theory on the above hyperkähler manifolds plays the central role.

    I would finally highlight three problems which penetrated my research from many directions:

    1. This is the quest to find the "right" compactification of my favourite spaces, e.g. whose intersection cohomology would calculate the Hodge cohomology of the original space.

    2. Is there a Hodge theoretical L^2-index interpretation of a certain signature, which can be defined on circle compact manifolds using the integration technique in my recent paper with Proudfoot? I conjecture that for a hyper-compact hyperkähler manifold this signature is always non-positive if the quaternionic dimension is odd and is non-negative if the quaternionic dimension is even; a result which is known for the L^2-signature by a result of Hitchin. My conjecture is known to be true for toric hyperkähler manifolds from a result in matroid theory.

    3. Understand if a strong version of the Hard Lefschetz theorem could be proven in a large class of these non-compact hyperkähler manifolds; a version of which is recently conjectured for the moduli space of Higgs bundles, as an outcome of our work with Rodriguez-Villegas on the number theory of these varieties. A weak version of the Hard Lefschetz theorem has been proven in my paper with Sturmfels and in my paper on the quaternionic geometry of matroids.